Death reports are personal reports that you will get after killing a monster. It will show you how your damage is divided between different damage types and division of damage between party members. In addition it shows duration of the kill and exp value. You can set this feature on by with 'set damage_report on' (unset to set it off).

Damage reports are very similar but for party. They accumulate from every kill the party makes. You can see them with 'party damages'. Reseting the party also resets this statistic. You do not need to have the personal death reports on to get the party version.